Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C

Average Error: 27.2 → 0.6

Time: 8.6s

Precision: binary64

Math

\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)\\ t_1 := \frac{z}{t_0}\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+72}:\\ \;\;\;\;\left(x + -2\right) \cdot \mathsf{fma}\left(\frac{4.16438922228}{x}, x, t_1\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+60}:\\ \;\;\;\;\left(x + -2\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{t_0}, x, t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3655.1204654076414}{x} + \left(\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) - \frac{130977.50649958357 - y}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (*
   (- x 2.0)
   (+
    (*
     (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y)
     x)
    z))
  (+
   (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x)
   47.066876606)))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          x
          (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
          47.066876606))
        (t_1 (/ z t_0)))
   (if (<= x -4.1e+72)
     (* (+ x -2.0) (fma (/ 4.16438922228 x) x t_1))
     (if (<= x 3.1e+60)
       (*
        (+ x -2.0)
        (fma
         (/
          (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
          t_0)
         x
         t_1))
       (+
        (/ 3655.1204654076414 x)
        (-
         (fma x 4.16438922228 -110.1139242984811)
         (/ (- 130977.50649958357 y) (* x x))))))))

C

double code(double x, double y, double z) {
	return ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
}

↓

double code(double x, double y, double z) {
	double t_0 = fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606);
	double t_1 = z / t_0;
	double tmp;
	if (x <= -4.1e+72) {
		tmp = (x + -2.0) * fma((4.16438922228 / x), x, t_1);
	} else if (x <= 3.1e+60) {
		tmp = (x + -2.0) * fma((fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y) / t_0), x, t_1);
	} else {
		tmp = (3655.1204654076414 / x) + (fma(x, 4.16438922228, -110.1139242984811) - ((130977.50649958357 - y) / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - 2.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606))
end

↓

function code(x, y, z)
	t_0 = fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)
	t_1 = Float64(z / t_0)
	tmp = 0.0
	if (x <= -4.1e+72)
		tmp = Float64(Float64(x + -2.0) * fma(Float64(4.16438922228 / x), x, t_1));
	elseif (x <= 3.1e+60)
		tmp = Float64(Float64(x + -2.0) * fma(Float64(fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y) / t_0), x, t_1));
	else
		tmp = Float64(Float64(3655.1204654076414 / x) + Float64(fma(x, 4.16438922228, -110.1139242984811) - Float64(Float64(130977.50649958357 - y) / Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision] * x), $MachinePrecision] + 137.519416416), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(x + 43.3400022514), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision] + 313.399215894), $MachinePrecision] * x), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]}, Block[{t$95$1 = N[(z / t$95$0), $MachinePrecision]}, If[LessEqual[x, -4.1e+72], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(4.16438922228 / x), $MachinePrecision] * x + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e+60], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] / t$95$0), $MachinePrecision] * x + t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(3655.1204654076414 / x), $MachinePrecision] + N[(N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision] - N[(N[(130977.50649958357 - y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.2
Target	0.8
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;x < -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x < 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -4.09999999999999963e72

   1. Initial program 64.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   2. Simplified 62.9

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]

   3. Taylor expanded in z around 0 62.9

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\left(\frac{\left(\left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x + y\right) \cdot x}{\left(313.399215894 + \left(263.505074721 + x \cdot \left(43.3400022514 + x\right)\right) \cdot x\right) \cdot x + 47.066876606} + \frac{z}{\left(313.399215894 + \left(263.505074721 + x \cdot \left(43.3400022514 + x\right)\right) \cdot x\right) \cdot x + 47.066876606}\right)} \]

   4. Simplified 62.6

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)} \]

   5. Applied egg-rr 62.6

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x + -2\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}\right)}^{3}} \]

   6. Taylor expanded in x around inf 1.8

      \[\leadsto {\left(\sqrt[3]{\left(x + -2\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{4.16438922228}{x}}, x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}\right)}^{3} \]

   7. Applied egg-rr 1.1

      \[\leadsto \color{blue}{1 \cdot \left(\left(x + -2\right) \cdot \mathsf{fma}\left(\frac{4.16438922228}{x}, x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)\right)} \]

   if -4.09999999999999963e72 < x < 3.1000000000000001e60

   1. Initial program 3.0

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   2. Simplified 0.9

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]

   3. Taylor expanded in z around 0 0.9

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\left(\frac{\left(\left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x + y\right) \cdot x}{\left(313.399215894 + \left(263.505074721 + x \cdot \left(43.3400022514 + x\right)\right) \cdot x\right) \cdot x + 47.066876606} + \frac{z}{\left(313.399215894 + \left(263.505074721 + x \cdot \left(43.3400022514 + x\right)\right) \cdot x\right) \cdot x + 47.066876606}\right)} \]

   4. Simplified 0.3

      \[\leadsto \left(x + -2\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)} \]

   if 3.1000000000000001e60 < x

   1. Initial program 63.8

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   2. Simplified 60.3

      \[\leadsto \color{blue}{\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}} \]

   3. Taylor expanded in x around inf 0.9

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + \left(4.16438922228 \cdot x + 3655.1204654076414 \cdot \frac{1}{x}\right)\right) - \left(130977.50649958357 \cdot \frac{1}{{x}^{2}} + 110.1139242984811\right)} \]

   4. Simplified 0.9

      \[\leadsto \color{blue}{\frac{3655.1204654076414}{x} - \left(\frac{130977.50649958357 - y}{x \cdot x} - \mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+72}:\\ \;\;\;\;\left(x + -2\right) \cdot \mathsf{fma}\left(\frac{4.16438922228}{x}, x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+60}:\\ \;\;\;\;\left(x + -2\right) \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, x, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3655.1204654076414}{x} + \left(\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right) - \frac{130977.50649958357 - y}{x \cdot x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1.0) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))